Steven Givant

Tarski's system of geometry

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhauser around 1978. It contains extended remarks about Tarskis system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

The Calculus of Relations as a Foundation for Mathematics

A variable-free, equational logic

Groups and Algebras of Binary Relations

\mathcal{L}^\times

Universal Classes of Simple Relation Algebras

based on the calculus of relations (a theory of binary relations developed by De Morgan, Peirce, and Schröder during the period 1864–1895) is shown to provide an adequate framework for the development of all of mathematics. The expressive and deductive powers of

The lattice of varieties of representable relation algebras

\mathcal{L}^\times

A representation theorem for measurable relation algebras

are equivalent to those of a system of first-order logic with just three variables. Therefore, three-variable first-order logic also provides an adequate framework for mathematics. Finally, it is shown that a variant of

Relation algebras and groups

\mathcal{L}^\times

Coset relation algebras

may be viewed as a subsystem of sentential logic. Hence, there are subsystems of sentential logic that are adequate to the task of formalizing mathematics.

Two-Quasi-Bijective Relation Algebras

In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras . He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jonsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarskis question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarskis system. In this paper we describe a far-reaching generalization of the positive results of Jonsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarskis axioms—called coset relation algebras —that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well. We prove that every atomic relation algebra satisfying a certain measurability condition—a condition generalizing the conditions imposed by Jonsson and Tarski—is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarskis problem by using coset relation algebras instead of the standard algebras of binary relations.

Varieties and universal classes

Tarski [19] proved the important theorem that the class of representable relation algebras is equationally axiomatizable. One of the key steps in his proof is showing that the class of (isomorphs of) simple set relation algebras-that is, algebras of binary relations with a unit of the form U x U for some non-empty set U-is universal, i.e., is axiomatizable by a set of universal sentences. In the same paper Tarski observed that the class of (isomorphs of) relation algebras constructed from groups (so-called group relation algebras) is also universal. We shall abstract the essential ingredients of Tarskis method (in Corollary 2.4), and then combine them with some observations about atom structures, to establish (in Theorem 2.6) a rather general method for showing that certain classes of simple relation algebras-and, more generally, certain classes of simple algebras in a discriminator variety V-are universal, and consequently that the collections of (isomorphs of) subdirect products of algebras in such classes form subvarieties of V. As applications of the method we show that two well-known classes of simple relation algebras, those constructed from projective geometries (sometimes called Lyndon algebras) and those constructed from modular lattices with a zero (sometimes called Maddux algebras), are universal. In the process we prove that these two classes consist precisely of all (isomorphs of) complex algebras over the respective geometries and modular lattices, provided that we choose the primitive notions of the latter structures in an appropriate fashion. We also derive Tarskis theorems and a related theorem of the author as easy corollaries of Theorem 2.6. Applications of the method to other domains of algebraic logic, such as cylindric and polyadic algebras, are similar in character.